# An explicit formula of Cauchy--Szeg\"{o} kernel for quaternionic Siegel upper half space and applications

## TL;DR

This paper derives an explicit formula for the Cauchy--Szeg"{o} kernel on quaternionic Siegel upper half space and explores its boundedness, compactness, and related operator properties on quaternionic Heisenberg groups.

## Contribution

It provides the first explicit formula for the kernel and establishes its Calderón--Zygmund properties, enabling analysis of associated operators.

## Key findings

- Kernel formula derived explicitly
- Cauchy--Szeg"{o} projection shown to be Calderón--Zygmund
- Characterizations of boundedness and compactness via BMO and VMO spaces

## Abstract

In this paper we obtain an explicit formula of Cauchy--Szeg\"{o} kernel for quaternionic Siegel upper half space, and then based on this, we prove that the Cauchy--Szeg\"{o} projection on quaternionic Heisenberg group is a Calder\'on--Zygmund operator via verifying the size and regularity conditions for the kernel. Next, we also obtain a suitable version of pointwise lower bound for the kernel, which further implies the characterisations of the boundedness and compactness of commutator of the Cauchy--Szeg\"{o} operator via the BMO and VMO spaces on quaternionic Heisenberg group, respectively.